
We would like to stress that this solution describes a BH only in the case where \(\varepsilon =1\)  and \(\varepsilon \eta <0\)  (see [1]}, [2]} for further details). Here, \(\mu \)  is an integration constant that plays the role of mass and \(l_\eta =\sqrt{|\varepsilon \eta |}\)  is a characteristic scale of the non-minimal coupling. The inverse tangent function restricts the domain of \(r\)  to \(r\in (0,\infty )\) . In the limit \(r\rightarrow 0\)  the function \(f(r)\)  yields Schwarzschild asymptotics, i.e. \(f(r)\approx 1-\frac{2\mu }{r}\) , while for \(r\rightarrow \infty \)  one obtains \(f(r)\approx \frac{3}{4}+\frac{r^2}{12 l^2_\eta }\)  i.e. AdS-like asymptotics. Note that in the metric functions (REF ) and () the non-minimal coupling \(l_\eta \)  is present, which is the strength of the coupling of the scalar field to curvature. Therefore, the BH solution is dressed with a scalar field given in Eq. ().
